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Pythagóras was an influential 6th-century BCE philosopher, but his ideas were not recorded during his lifetime, and most of what is known about him and his followers came from much later sources, especially Socrates’ students a century and a half later, who attributed his ideas to the Pythagoreans rather than to Pythagóras himself, and the Neoplatonists, who were active from the 3rd century until emperor Flávios Pétros Sabbátios Ioustinianós [“Justinian”] closed the Platonic Academy in Athens in 529. Allegedly the first to call himself a philosopher (lover of wisdom), he was born on the island of Samos, a particularly rich and powerful city-state in the eastern Aegean Sea, renowned for its vineyards (the source of muscat wine) and red pottery. According to Strabo, the name Samos came from the Phoenician term for "rise by the shore." Though it was later said that he was the son of Apollo, his father was actually a gem-engraver or merchant from Tyre. Socrates’ student Aristippus, the founder of the Cyrenaic school, explained his name by saying, "He spoke (agor-) the truth no less than did the Pythian (Pyth-)," the oracle of Delphi, and in the 3rd century Iamblichus Chalcidensis claimed that the Pythia herself had told his mother that she would bear a son who would be supremely beautiful, wise, and beneficial to humankind. (Another late source gave his mother's name as Pythais).

Iamblichus summarized his early life thus: "Twenty-two years Pythagoras remained in Egypt, pursuing closely his investigations, visiting every place famous for its teachings, every person celebrated for wisdom. Astronomy and geometry he especially studied and he was thoroughly initiated in all the mysteries of the gods, till, having been taken captive by the soldiers of Cambyses, he was carried to Babylon. Here the Magi instructed him in their venerable knowledge and he arrived at the summit of arithmetic, music and other disciplines. After twelve years he returned to Samos, being then about fifty-six years of age." He may have also visited India but returned to Samos in ca. 520 BCE; about 30 years later he moved to Kroton in Calabria, renowned for its physicians, and established a disciplined, tightly-organized, secretive school or guild. According to Diogenes Laërtius, his followers kept their doctrines secret, tended to ascribe all of their findings to him personally, and practiced temperance of all kinds, possibly including vegetarianism. Some authors mentioned a "Pythagorean diet," the abstention from eating meat, beans, or fish. The aversion to beans probably had a magico-religious reason, since they resemble the kidneys and genitalia, and it was believed that beans and humans were created from the same material. According to Diogenes Laertius, fava beans were particularly sacred because they have hollow stems; the Pythagoreans believed that the souls of the dead would travel through the ground, up the stems, and into the beans, where they would reside. (According to one account of Pythagoras’ death, his enemies set fire to his house, forcing him to flee toward a bean field, where he halted and declared he would rather die than enter it – whereupon his pursuers slit his throat.) The medical theorist Alcmaeon was among his students, and also his future wife Theano, who told Hippodamus of Miletus that her treatise “On Virtue” contained the doctrine of the golden mean. She denied that “Pythagoras said all things are generated from number. The very assertion poses a difficulty: How can things which do not exist even be conceived to generate?... Rather, in accordance with number -- on the grounds that order in the primary sense is in number and it is by participation in order that a first and a second and the rest sequentially are assigned to things which are counted." One of their sons, Arignote, composed a dictum among the “Sacred Discourses”: "The eternal essence of number is the most providential cause of the whole heaven, earth and the region in between. Likewise it is the root of the continued existence of the gods and daimones, as well as that of divine men." The school acquired considerable influence with Kroton’s ruling council of 1,000. In 510 BCE the city sent 100,000 men under Pythagoras’ son-in-law, the wrestler Milo, against its rival Sybaris and destroyed it. Shortly afterwards the Pythagoreans led the opposition to a democratic constitution, but Cylon (who had been blackballed by them) and Ninon led a successful popular revolution. Milo’s house was attacked and set on fire. Some sources claimed that Pythagoras perished with most of his disciples, others that he fled to Tarentum; driven from there, he escaped to Metapontum and fasted to death. His followers were suppressed everywhere as organized groups, but occasionally an individual rose to prominence.

According to Nicomachus, Pythagoras’ successor as head of the school was Philolaus, who was born either in Kroton, Tarentum, or Metapontum. He may have been the first to commit Pythagoreanism to writing, in a volume that seems to have had a profound effect on Plato, especially in the development of the ideas espoused in “Timaeus.” The first part contained a general account of the origin and arrangement of the universe, including the notion that the foundation of everything is the part played by the harmonious combinations of the limiting and the limitless (from which time, space, and motion were derived). The second part was an exposition of the nature of numbers, the “dominant and self-produced bond of the eternal continuance of things." But number has two forms, the even and the odd, and a third, the even-odd that results from the mixture of the two. But one, or unity, is the essence of number, the origin of all numbers, and so of all things; and the original unity is God. Philolaus was the teacher of Plato’s friend Archytas of Tarentum, the founder of mathematical mechanics; he named (but did not invent) the harmonic mean, which became important in projective geometry and number theory; 500 years later, Aulus Gellius reported that he built a self-propelled, bird-shaped flying device. Based on Pythagorean ideas, Polykleitos of Argos, a contemporary of Phidias, created the Classical Greek style. In the “Kanon,” he wrote that beauty consists in the proportion of the parts rather than the elements (materials). "Perfection comes about little by little through many numbers.” The method treats some part, such as the distal phalange of the little finger, as one side of a square; rotating that square's diagonal gives a rectangular proportion suitable for the medial phalange; the method is repeated for each phalange; then the whole finger becomes the basis for the palm; the whole hand begets the lower forearm, which begets the upper arm; and so on. Then, to demonstrate the thesis, he designed a male nude, also known as “Kanon.” Pythagorean lawmakers were held in particular esteem, including Charondas of Catania in Sicily, whose laws were written in verse and adopted by the Chalcidic colonies in Sicily and Italy; penalties for perjury were their innovation, along with their clarity and precision.

Many myths about Pythagoras circulated: that he gleamed with a supernatural brightness; that he had a golden thigh; that he could be at more than one place simultaneously; that Abaris the Hyperborean, a sage, priest of Apollo, and healer from the time of Croesus, who purified Sparta and Knossos from plague, who ate no food and traveled around the world on a golden arrow, who (with Pythagoras) tried to instill virtue in the tyrant Phalaris of Acragas (Agrigento, Sicily), a notorious cannibal who executed his enemies in a brazen bull invented by Perillos of Athens: the victims were shut in and roasted alive; he was overthrown by Telemachus and burned in his own brazen bull. Another tale had Pythagoras visit the Cave of Ida with the seer and prophet-poet Epimenides of Knossos, one of the founders of Orphism, who slept for 57 years in the Cave of Ida and awoke with the gift of prophecy, purged Athens, assisted Solon in his reforms, made military predictions in Sparta, and was captured in a war between the Spartans and Cnossians and executed because he refused to prophesy favorably for his captors; in another version, when he died in Crete at nearly 200, it was found that his skin was covered with tattooed writing, which was then preserved at Sparta; Loukãs (“Luke”) quoted him in the “Acts of the Apostles” [17:28], and St. Paul also, when he warned Titus [1:12] about the Cretans.

Aristoxenus (4th century BCE) claimed that Themistokleia, a priestess at Delphi, taught Pythagoras his moral doctrines (the 10th-century “Suda” encyclopedia called her Theokleia and claimed she was his sister.) The 1st-century historian Plutarchus asserted that Pythagoras received instruction from the Egyptian priest Oenuphis of Heliopolis, while the 2nd-century Christian theologian Titus Flavius Clemens (“Clement of Alexandria”) held that he was a disciple of the Egyptian archprophet Soches, and the Muslims claimed he had been initiated by Thoth (Hermes). In his travels among the Greeks – Delos, Sparta, Phlius, Crete, etc. -- he was usually described in a religious or priestly role or as a lawgiver, and he was said to have practiced divination. Many mathematical and scientific discoveries were attributed to him, including (in the 4th century) the “Pythagorean theorem” (in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides), but Babylonians understood the formula, and Indian and Chinese mathematicians also discovered the theorem in advance of Pythagoras. He translated musical notes into mathematical equations and discovered the properties of string length. The diatonic ("Pythagorean") scale is such that the ratio of the highest to the lowest pitch is 2:1, which produces an octave, which in turn is divided into a fifth and a fourth, which have the ratios of 3:2 and 4:3 respectively and which, when multiplied, also make an octave. Going up a fifth from the lowest note in the octave and then up a fourth from there produces the octave’s highest note. The fifth is the multiplication of the three largest whole tones (each corresponding to the ratio of 9:8) plus a perfecting semitone. Likewise, the fourth can be made up of two whole tones and the same perfecting semitone. The Pythagorean theory of harmonic ratios continued to form the basis of music theory in the Muslim world, such as al-Farabi's “Kitab al-Musiqa al-kabir.” Using mixtures of diatonic, chromatic and enharmonic melodies, Pythagoras developed a technique to heal disease and emotional turmoil. He promoted the idea of the "harmony of the spheres" that posited the movement of planets according to mathematical equations which corresponded to musical notes and thus produced a symphony. He may have been the first to suggest that the Earth is spherical, with antipodes, and that it revolves around the sun, which is a movable sphere in the center of the universe, around which the planets revolve.

He devised the tetractys, a triangular figure of four rows which add up to the perfect number, ten; as a mystical symbol, it was very important to the Pythagoreans, who swore oaths by it: “By him who handed to our generation the tetractys, source of the roots of ever-flowing nature." He suggested that he could write something in blood on a mirror and project the message onto the moon. Many authorities noted the similarities between his religious and ascetic principles with the Orphic or Cretan mysteries. Xenophanes wrote that he believed in the transmigration of souls and related the story of Pythagoras interceding on behalf of a dog that was being beaten, protesting that he recognized the cries of a departed friend. Philostratus wrote that Pythagoras knew not only who he was but who he had been; Heraclides Ponticus said Pythagoras claimed that he had lived four previous lives that he could remember in detail, including as Euphorbus, the Trojan who wounded Patroclus before he was killed by Hector and who was then slain by Menelaus; to prove it, he went to the temple of Hera in Argos and immediately identified the shield of Euphorbus; he had been Mercury’s son AEthalides, and Hermotimos, the prophet of Clazomenae. Aulus Gellius said Pythagoras was the reincarnation of a beautiful courtesan; he has also been a merchant and a fisherman. Iamblichus wrote that he tamed a bear and an eagle by “stroking it gently with his hand” and held absolute dominion over animals by “the power of his voice” or the “influence of his touch.” He taught that the soul has three vehicles: the ethereal, which is luminous and celestial, in which the soul resides in a state of bliss among the stars; the luminous, which suffers the punishment of sin after death; and the terrestrial, which is the vehicle it occupies on this earth. The ideas of Pythagoras and the Pythagoreans continued to exert influence long after the formal school disappeared, in large part through the writings of Plato and the Neoplatonists. But they clearly influenced the thought of later doctrines as well, such as hermeticism, gnosticism, alchemy, and numerology. Freemasonry and Rosicrucianism claim to have evolved out of the Pythagorean Brotherhood. Pythagorean philosophy had a marked impact on the scientific revolution, especially due to the focus given to the “Platonic solids,” which Plato derived from Pythagorean theories of geometry and numbers; in the “Timaeus,” he associated them with the four classical elements; as for the fifth solid, “the god used [it] for arranging the constellations on the whole heaven." In three-dimensional space, a Platonic solid is a regular, convex polyhedron, constructed by congruent regular polygonal faces with the same number of faces meeting at each vertex. Only five solids meet those criteria: tetrahedrons, cubes, octahedrons, dodecahedrons, and icosahedrons. (Proclus credited Pythagoras with their discovery, but he may have only been familiar with three of them since Plato’s contemporary, Theaetetus, seems to have discovered the octahedron and icosahedron.) In the 16th century, Johannes Kepler fruitlessly tried to relate the five known extraterrestrial planets to the five Platonic solids, but out of that failed effort came his three laws of orbital dynamics, the first of which was that the orbits of planets are ellipses rather than circles, thus changing the course of physics and astronomy. In the 20th century, Robert Moon tried to link the Platonic solids in the physical world to the electron shell model in chemistry, in a theory known as the "Moon model."